3.599 \(\int x^2 (a+b x^3)^p \, dx\)

Optimal. Leaf size=23 \[ \frac {\left (a+b x^3\right )^{p+1}}{3 b (p+1)} \]

[Out]

1/3*(b*x^3+a)^(1+p)/b/(1+p)

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {261} \[ \frac {\left (a+b x^3\right )^{p+1}}{3 b (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^p,x]

[Out]

(a + b*x^3)^(1 + p)/(3*b*(1 + p))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x^2 \left (a+b x^3\right )^p \, dx &=\frac {\left (a+b x^3\right )^{1+p}}{3 b (1+p)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\left (a+b x^3\right )^{p+1}}{3 b (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^p,x]

[Out]

(a + b*x^3)^(1 + p)/(3*b*(1 + p))

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fricas [A]  time = 0.63, size = 25, normalized size = 1.09 \[ \frac {{\left (b x^{3} + a\right )} {\left (b x^{3} + a\right )}^{p}}{3 \, {\left (b p + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^p,x, algorithm="fricas")

[Out]

1/3*(b*x^3 + a)*(b*x^3 + a)^p/(b*p + b)

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giac [A]  time = 0.15, size = 21, normalized size = 0.91 \[ \frac {{\left (b x^{3} + a\right )}^{p + 1}}{3 \, b {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^p,x, algorithm="giac")

[Out]

1/3*(b*x^3 + a)^(p + 1)/(b*(p + 1))

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maple [A]  time = 0.00, size = 22, normalized size = 0.96 \[ \frac {\left (b \,x^{3}+a \right )^{p +1}}{3 \left (p +1\right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)^p,x)

[Out]

1/3*(b*x^3+a)^(p+1)/b/(p+1)

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maxima [A]  time = 1.36, size = 21, normalized size = 0.91 \[ \frac {{\left (b x^{3} + a\right )}^{p + 1}}{3 \, b {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^p,x, algorithm="maxima")

[Out]

1/3*(b*x^3 + a)^(p + 1)/(b*(p + 1))

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mupad [B]  time = 1.09, size = 21, normalized size = 0.91 \[ \frac {{\left (b\,x^3+a\right )}^{p+1}}{3\,b\,\left (p+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^3)^p,x)

[Out]

(a + b*x^3)^(p + 1)/(3*b*(p + 1))

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sympy [A]  time = 2.13, size = 134, normalized size = 5.83 \[ \begin {cases} \frac {x^{3}}{3 a} & \text {for}\: b = 0 \wedge p = -1 \\\frac {a^{p} x^{3}}{3} & \text {for}\: b = 0 \\\frac {\log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + x \right )}}{3 b} + \frac {\log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac {1}{b}} + 4 x^{2} \right )}}{3 b} & \text {for}\: p = -1 \\\frac {a \left (a + b x^{3}\right )^{p}}{3 b p + 3 b} + \frac {b x^{3} \left (a + b x^{3}\right )^{p}}{3 b p + 3 b} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**3+a)**p,x)

[Out]

Piecewise((x**3/(3*a), Eq(b, 0) & Eq(p, -1)), (a**p*x**3/3, Eq(b, 0)), (log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3)
+ x)/(3*b) + log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(3*b),
Eq(p, -1)), (a*(a + b*x**3)**p/(3*b*p + 3*b) + b*x**3*(a + b*x**3)**p/(3*b*p + 3*b), True))

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